Abstract:
Guy asks if there exists a point in the plane at rational distance to the corners of
the unit square. Also known as the four-distance problem, we establish the equivalence
of the problem to the existence of nontrivial solutions to a particular Pythagorean triple,
from which we derive known conditions and establish new results. We then provide a
generalization given by Barbara of the four-distance problem to regular polygons of unit
side, in which a negative answer is almost always obtained.
